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\title{Adaptive vorticity confinement for finite volume method}
\author{The Author}
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\begin{equation*}
U_t + f_x + g_y = 0
\end{equation*}

\begin{equation*}
\dd{U_{i,j}}{t} + \frac{F_{i+1/2,j} - F_{i-1/2,j}}{\Delta x} + \frac{G_{i,j+1/2} - G_{i,j-1/2}}{\Delta y} = 0
\end{equation*}

\begin{eqnarray*}
F_{i+1/2,j} = F(U_{i+1/2,j}^-, U_{i+1/2,j}^+) &=& \frac{1}{2}[ f(U_{i+1/2,j}^-) + f(U_{i+1/2,j}^+)] + F_{d_{i+1/2,j}} \\
&=& \mu_x f_{i+1/2,j} + F_{d_{i+1/2,j}}
\end{eqnarray*}

\begin{equation*}
\frac{F_{i+1/2,j} - F_{i-1/2,j}}{\Delta x} = \delta_x \mu_x f_{i,j} + \delta_x F_{d_{i,j}}
\end{equation*}

\begin{equation*}
\dd{U}{t} + 
\delta_x \mu_x f + \delta_x F_{d} + \delta_y \mu_y g + \delta_y G_d = 0
\end{equation*}

Vorticity
\begin{equation*}
\Omega = (\rho v)_x - (\rho u)_y \approx \delta_x U^3 - \delta_y U^2
\end{equation*}

\begin{eqnarray*}
\dd{\Omega}{t} &+& \delta_x[ \delta_x \mu_x f^{(3)} + \delta_x F^{(3)}_{d} + \delta_y \mu_y g^{(3)} + \delta_y G^{(3)}_d] \\
&-& \delta_y[ \delta_x \mu_x f^{(2)} + \delta_x F^{(2)}_{d} + \delta_y \mu_y g^{(2)} + \delta_y G^{(2)}_d] = \delta_x \epsilon k^{(2)} - \delta_y \epsilon k^{(1)}
\end{eqnarray*}

To cancel the dissipation term, choose
\begin{eqnarray*}
\delta_x G_d^{(3)} - \delta_y G_d^{(2)} &=& -\epsilon k^{(1)} \\
\delta_x F_d^{(3)} - \delta_y F_d^{(2)} &=& +\epsilon k^{(2)}
\end{eqnarray*}

But $f^{(2)}$ and $g^{(3)}$ contain pressure, which should not appear in the vorticity equation. These spurious pressure terms are
\begin{equation*}
\delta_x \delta_y \mu_y p - \delta_y \delta_x \mu_x p
\end{equation*}

\end{document}  
